PANELS WITH NONSTATIONARY MULTIFACTOR ERROR STRUCTURES … · (1993), Conley and Dupor (2003), Conley an d Topa (2002) and Pesaran, Schuermann, and Weiner (2004), as well as the literature

PANELS WITH NONSTATIONARY MULTIFACTOR ERROR STRUCTURES

GEORGE KAPETANIOS M. HASHEM PESARAN TAKASHI YAMAGATA

CESIFO WORKING PAPER NO. 1788

CATEGORY 10: EMPIRICAL AND THEORETICAL METHODS AUGUST 2006

An electronic version of the paper may be downloaded • from the SSRN website: www.SSRN.com • from the RePEc website: www.RePEc.org

• from the CESifo website: Twww.CESifo-group.deT

CESifo Working Paper No. 1788

PANELS WITH NONSTATIONARY MULTIFACTOR ERROR STRUCTURES

Abstract The presence of cross-sectionally correlated error terms invalidates much inferential theory of panel data models. Recently work by Pesaran (2006) has suggested a method which makes use of cross-sectional averages to provide valid inference for stationary panel regressions with multifactor error structure. This paper extends this work and examines the important case where the unobserved common factors follow unit root processes and could be cointegrated. It is found that the presence of unit roots does not affect most theoretical results which continue to hold irrespective of the integration and the cointegration properties of the unobserved factors. This finding is further supported for small samples via an extensive Monte Carlo study. In particular, the results of the Monte Carlo study suggest that the cross-sectional average based method is robust to a wide variety of data generation processes and has lower biases than all of the alternative estimation methods considered in the paper.

JEL Code: C12, C13, C33.

Keywords: cross section dependence, large panels, unit roots, principal components, common correlated effects.

George Kapetanios Department of Economics

Queen Mary, University of London Mile End Road London E1 4NS United Kingdom

[email protected]

M. Hashem Pesaran Faculty of Economics Cambridge University Cambridge, CB3 9DD

United Kingdom [email protected]

Takashi Yamagata

Faculty of Economics Cambridge University Cambridge, CB3 9DD

United Kingdom [email protected]

July 31, 2006 The authors thank Vanessa Smith and Elisa Tosetti for helpful comments on a preliminary version of this paper. Hashem Pesaran and Takashi Yamagata acknowledge financial support from the ESRC (Grant No. RES-000-23-0135).

1 Introduction

Panel data sets have been increasingly used in economics to analyze complex economic

phenomena. One of their attractions is the ability to use an extended data set to obtain

information about parameters of interest which are assumed to have common values across

panel units. Most of the work carried out on panel data has usually assumed some form of

cross sectional independence to derive the theoretical properties of various inferential proce-

dures. However, such assumptions are often suspect and as a result recent advances in the

literature have focused on estimation of panel data models subject to error cross sectional

dependence.

A number of different approaches have been advanced for this purpose. In the case of

spatial data sets where a natural immutable distance measure is available the dependence

is often captured through “spatial lags” using techniques familiar from the time series lit-

erature. In economic applications, spatial techniques are often adapted using alternative

measures of “economic distance”. This approach is exemplified in work by Lee and Pesaran

(1993), Conley and Dupor (2003), Conley and Topa (2002) and Pesaran, Schuermann, and

Weiner (2004), as well as the literature on spatial econometrics recently surveyed by Anselin

(2001). In the case of panel data models where the cross section dimension (N) is small

(typically N < 10) and the time series dimension (T ) is large the standard approach is to

treat the equations from the different cross section units as a system of seemingly unrelated

regression equations (SURE) and then estimate the system by the Generalized Least Squares

(GLS) techniques.

In the case of panels with large cross section dimension, SURE approach is not practical

and has led a number of investigators to consider unobserved factor models, where the cross

section error correlations are defined in terms of the factor loadings. Use of factor models

is not new in economics and dates back to the pioneering work of Stone (1947) who applied

the principal components (PC) analysis of Hotelling to US macroeconomic time series over

the period 1922-1938 and was able to demonstrate that three factors (namely total income,

its rate of change and a time trend) explained over 97 per cent of the total variations of

all the 17 macro variables that he had considered. Until recently, subsequent applications

of the PC approach to economic times series has been primarily in finance. See, for ex-

ample, Chamberlain and Rothschild (1983), Connor and Korajzcyk (1986) and Connor and

Korajzcyk (1988). But more recently the unobserved factor models have gained popularity

for forecasting with a large number of variables as advocated by Stock and Watson (2002).

2

The factor model is used very much in the spirit of the original work by Stone, in order to

summarize the empirical content of a large number of macroeconomics variables by a small

set of factors which, when estimated using principal components, is then used for further

modelling and/or forecasting. A related literature on dynamic factor models has also been

put forward by Forni and Reichlin (1998) and Forni, Hallin, Lippi, and Reichlin (2000).

Recent uses of factor models in forecasting focuses on consistent estimation of unobserved

factors and their loadings. Related theoretical advances by Bai and Ng (2002) and Bai (2003)

are also concerned with estimation and selection of unobserved factors and do not consider

the estimation and inference problems in standard panel data models where the objects of

interest are slope coefficients of the conditioning variables (regressors). In such panels the

unobserved factors are viewed as nuisance variables, introduced primarily to model the cross

section dependencies of the error terms in a parsimonious manner relative to the SURE for-

mulation.

Despite these differences knowledge of factor models could still be useful for the analysis

of panel data models if it is believed that the errors might be cross sectionally correlated.

Disregarding the possible factor structure of the errors in panel data models can lead to in-

consistent parameter estimates and incorrect inference. Coakley, Fuertes, and Smith (2002)

suggest a possible solution to the problem using the method of Stock and Watson (2002).

But, as Pesaran (2006) shows, the PC approach proposed by Coakley, Fuertes, and Smith

(2002) can still yield inconsistent estimates. Pesaran (2006) suggests a new approach by

noting that linear combinations of the unobserved factors can be well approximated by cross

section averages of the dependent variable and the observed regressors. This leads to a new

set of estimators, referred to as the Common Correlated Effects estimators, that can be com-

puted by running standard panel regressions augmented with the cross section averages of

the dependent and independent variables. The CCE procedure is applicable to panels with

a single or multiple unobserved factors so long as the number of unobserved factors is fixed.

In this paper we extend the analysis of Pesaran (2006) to the case where one or more

of the unobserved common factors could be integrated of order 1, or I(1). This also allows

for the possibility of cointegration amongst the I(1) factors and does not require an a priori

knowledge of the number of unobserved factors, or their integration or cointegration prop-

erties. It is only required that the number of unobserved factors remain fixed as the sample

size is increased. The theoretical findings of the paper are further supported for small sam-

ples via an extensive Monte Carlo study. In particular, the results of the Monte Carlo study

3

clearly show that the CCE estimator is robust to a wide variety of data generation processes

and has lower biases than all of the alternative estimation methods considered in the paper.

The structure of the paper is as follows: Section 2 provides an overview of the method

suggested by Pesaran (2006) in the case of stationary factor processes. Section 3 provides

the theoretical framework of the analysis of nonstationarity. In this section the theoretical

properties of the various estimators are presented. Section 4 presents an extensive Monte

Carlo study, and Section 5 concludes.

2 Panel Data Models with Observed and UnobservedCommon Effects

In this section we review the methodology introduced in Pesaran (2006). Let yit be the

observation on the ith cross section unit at time t for i = 1, 2, ..., N ; t = 1, 2, ..., T, and

suppose that it is generated according to the following linear heterogeneous panel data model

yit = α0idt + β0ixit + γ

0ift + εit, (1)

where dt is a n × 1 vector of observed common effects, which is partitioned as dt =(d01t,d

02t,d

03t)

0 where d1t is a n1× 1 vector of deterministic components such as intercepts orseasonal dummies, d2t is a n2×1 vector of unit root stochastic observed common effects andd3t is a n3×1 vector of stationary stochastic observed common effects, with n = n1+n2+n3,

xit is a k× 1 vector of observed individual-specific regressors on the ith cross section unit attime t, ft is them×1 vector of unobserved common effects, and εit are the individual-specific(idiosyncratic) errors assumed to be independently distributed of (dt,xit). The unobserved

factors, ft, could be correlated with (dt,xit), and to allow for such a possibility the following

specification for the individual specific regressors will be considered

xit = A0idt + Γ0ift + vit, (2)

where Ai and Γi are n×k and m×k, factor loading matrices with fixed components, vit are

the specific components of xit distributed independently of the common effects and across

i, but assumed to follow general covariance stationary processes. In this paper we allow for

one or more of the common factors, dt and ft, to be I(1).

Combining (1) and (2) we now have

zit(k+1)×1

=

µyitxit

¶= B0i

(k+1)×ndtn×1

+ C0i(k+1)×m

ftm×1

+ uit(k+1)×1

, (3)

4

where

uit =

µεit + β

0ivit

vit

¶, (4)

Bi =¡αi Ai

¢µ In 0βi Ik

¶, Ci =

¡γi Γi

¢µ Im 0βi Ik

¶, (5)

Ik is an identity matrix of order k, and the rank of Ci is determined by the rank of the

m× (k + 1) matrix of the unobserved factor loadings

Γi =¡γi Γi

¢. (6)

As discussed in Pesaran (2006), the above set up is sufficiently general and renders a variety

of panel data models as special cases. In the panel literature with T small and N large, the

primary parameters of interest are the means of the individual specific slope coefficients, βi,

i = 1, 2, ..., N . The common factor loadings, αi and γi, are generally treated as nuisance

parameters. In cases where both N and T are large, it is also possible to consider consistent

estimation of the factor loadings. The presence of unobserved factors in (1) implies that

estimation of βi and its cross sectional mean cannot be undertaken using standard methods.

Pesaran (2006) has suggested using cross section averages of yit and xit as proxies for the

unobserved factors in (1). To see why such an approach could work, consider simple cross

section averages of the equations in (3)1

zt = B0dt + C0ft + ut, (7)

where

zt =1

N

NXi=1

zit, ut =1

N

NXi=1

uit,

and

B =1

N

NXi=1

Bi, C =1

N

NXi=1

Ci. (8)

Suppose that

Rank(C) = m ≤ k + 1, for all N. (9)

Then, we have

ft =³CC

0´−1C¡zt − B0dt − ut

¢. (10)

But since

utq.m.→ 0, as N →∞, for each t, (11)

and

Cp→ C = Γ

µIm 0β Ik

¶, as N →∞, (12)

1Pesaran (2006) considers cross section weighted averages that are more general. But to simplify theexposition we confine our discussion to simple averages throughout.

5

where

Γ = (E (γi) , E (Γi)) = (γ,Γ). (13)

It follows, assuming that Rank(Γ) = m, that

ft − (CC0)−1C¡zt − B0dt

¢ p→ 0, as N →∞.

This suggests using ht = (d0t, z0t)0 as observable proxies for ft, and is the basic insight that

lies behind the Common Correlated Effects estimators developed in Pesaran (2006). It is

further shown that the CCE estimation procedure in fact holds even if Γ turns out to be

rank deficient.

We now discuss the two estimators for the means of the individual specific slope coef-

ficients proposed by Pesaran (2006). One is the Mean Group (MG) estimator proposed in

Pesaran and Smith (1995) and the other is a generalization of the fixed effects estimator that

allows for the possibility of cross section dependence. The former is referred to as the “Com-

mon Correlated Effects Mean Group” (CCEMG) estimator, and the latter as the “Common

Correlated Effects Pooled” (CCEP) estimator.

The CCEMG estimator is a simple average of the individual CCE estimators, bi of βi,

bMG = N−1NXi=1

bi. (14)

where

bi = (X0iMXi)

−1X0iMyi, (15)

Xi = (xi1,xi2, ...,xiT )0, yi = (yi1, yi2, ..., yiT )0, M is defined by

M = IT − H¡H0H

¢−1H0, (16)

H = (D, Z), D and Z being, respectively, the T ×n and T × (k+1) matrices of observationson dt and zt.

Efficiency gains from pooling of observations over the cross section units can be achieved

when the individual slope coefficients, βi, are the same. Such a pooled estimator of β,

denoted by CCEP, is given by

bP =

ÃNXi=1

X0iMXi

!−1 NXi=1

X0iMyi. (17)

6

3 Theoretical Properties of CCE Estimators in Non-stationary Panel Data Models

The following assumptions will be used in the derivation of the asymptotic properties of the

CCE estimators.

Assumption 1 (non-stationary common effects): The (n2+m)× 1 vector of stochasticcommon effects, gt = (d02t, f

0t)0, is a multivariate unit root process given by

gt = gt−1 + ζgt

where ζgt is a (n2+m)× 1 vector of L2 stationary near epoque dependent (NED) processesof size 1/2, distributed independently of the individual-specific errors, εit0 and vit0 for all i,

t and t0.

Assumption 2 (stationary common effects): The n3 × 1 vector of common effects, d3t,is a covariance stationary process given by

d3t =∞X=0

J ψt− , (18)

where the matrices J satisfy the condition

∞X=0

s||J ||<∞, (19)

for some s ≥ 1/2, and the ψt are distributed independently of the individual-specific errors,

εit0 and vit0 for all i, t and t0.

Assumption 3 (individual-specific errors): The individual specific errors εit and vjtare distributed independently for all i, j and t. For each i, εit is serially uncorrelated with

mean zero, a finite variance σ2i < K, and a finite fourth-order cumulant. vit follows a linear

stationary process with absolute summable autocovariances given by

vit =∞X=0

Si νi,t− , (20)

where νit are k × 1 vectors of identically, independently distributed (IID) random variables

with mean zero, the variance matrix, Ik, and finite fourth-order cumulants. In particular,

the k × k coefficient matrices Si satisfy the condition

∞X=0

s||Si ||<∞, (21)

for all i and some s ≥ 1/2.

7

Assumption 4 (factor loadings): The unobserved factor loadings, γi and Γi, are in-

dependently and identically distributed across i, and of the individual specific errors, εjt

and vjt, the common factors, gt = (d02t, f0t), for all i, j and t with fixed means γ and Γ,

respectively, and finite variances. In particular,

γi = γ + ηi, ηi v IID (0,Ωη), for i = 1, 2, ..., N, (22)

where Ωη is a m×m symmetric non-negative definite matrix, and kγk < K, kΓk < K, and

kΩηk < K.

Assumption 5 (random slope coefficients): The slope coefficients, βi, follow the random

coefficient model

βi = β + υi, υi v IID (0,Ωυ), for i = 1, 2, ..., N, (23)

where kβk < K, kΩυk < K, Ωυ is a k× k symmetric non-negative definite matrix, and the

random deviations, υi, are distributed independently of γj,Γj,εjt, vjt, and gt for all i, j and

t.

Assumption 6: (identification of βi and β): Consider the cross section averages of the

individual specific variables, zit, defined by zt = 1N

PNj=1 zjt, and let

M = IT − H¡H0H

¢−H0, (24)

and

Mg = IT −G (G0G)−G0, (25)

whereG = (D,F),D =(d1,d2, ...,dT )0, F =(f1, f2, ..., fT )

0 are T×n and T×m data matrices

on observed and unobserved common factors, respectively, Z = (z1, z2, ..., zT )0 is the T×(k+1) matrix of observations on the cross section averages, and

¡H0H

¢−and (G0G)− denote

the generalized inverses of H0H and G0G, respectively. Also denote the T × k observation

matrix on individual specific regressors by Xi = (xi1,xi2, ...,xiT )0.

6a: (identification of βi): The k×kmatrices ΨiT = T−1¡X0

iMXi

¢andΨig = T−1 (X0

iMgXi)

are non-singular and Ψ−1iT and Ψ

−1ig have finite second order moments, for all i.

6b: (identification of β): The k × k pooled observation matrix ΨNT defined by

ΨNT =1

N

NXi=1

µX0

iMXi

T

¶(26)

is non-singular.

Remark 1 Note that Assumption 3 is slightly stronger than Assumption 2 of Pesaran (2006)

which requires that

V ar (vit) =∞X=0

Si S0i = Σi ≤ K <∞, (27)

8

for all i and some constant matrixK, where Σi is a positive definite matrix, rather than (21).

Further note that (21) implies that vit are L2 stationary near epoque dependent processes of

size 1/2.

Remark 2 Assumption 1 implicitly allows for cointegration among the m unobserved fac-

tors. To see this it is useful to impose some further structure on ζ2gt where ζgt = (ζ01gt, ζ

02gt)

0

is a partition of ζgt comformable to the partition of gt in terms of d2t and ft. If we let

ζ2gt =P∞

=0W t− where t− are i.i.d. with finite variance andP∞

=0 ||W || < ∞ then

ifP∞

=0W is of reduced rank equal to r, then ζ2gt exhibits cointegration. Further ft may be

represented by a set of r stationary components and m− r random walk components. This

representation will be of use in the proofs of our results. Note that the above MA represen-

tation of ζ2gt implies that ζ2gt is a vector of L2 stationary near epoque dependent (NED)

processes of size 1 and hence satisfy assumption 12.

For each i and t = 1, 2, ..., T , writing the model in matrix notation we have

yi =Dαi +Xiβi + Fγi + εi, (28)

where εi = (εi1, εi2, ..., εiT )0, and as set out in Assumption 5, D = (d1,d2, ...,dT )

0 and

F = (f1, f2, ..., fT )0. Using (28) in (15) we have

bi − βi =

µX0

iMXi

T

¶−1µX0

iMF

T

¶γi+

µX0

iMXi

T

¶−1µX0

iMεiT

¶, (29)

which shows the direct dependence of bi on the unobserved factors through T−1X0iMF. To

examine the properties of this component, writing (2) and (7) in matrix notations, we first

note that

Xi = GΠi +Vi, (30)

and

H = GP+ U∗, (31)

where Πi = (A0i,Γ

0i)0, Vi = (vi1,vi2, ...,viT )

0 ,

P(n+m)×(n+k+1)

=

µIn B0 C

¶, U∗ = (0, U), (32)

Using Lemma 1 which forms the basis of all the theoretical results of this paper and

assuming that the rank condition (9) is satisfied, it follows that

X0iMF

T=X0

iMgF

T+Op

µ1√NT

¶+Op

µ1

N

¶, (33)

2Our analysis allows for cointegration in d2t as well. However, for the sake of simplicity we do notexplicitly analyse this case which is a straightforward extension of the results presented in this paper.

9

X0iMXi

T=X0

iMgXi

T+Op

µ1√NT

¶+Op

µ1

N

¶, (34)

andX0

iMεiT

=X0

iMgεiT

+Op

µ1√NT

¶+Op

µ1

N

¶. (35)

Note that F ⊂ G and hence that T−1X0iMgF = 0. If the rank condition does not hold

then again by Lemma 1 it follows that

X0iMF

T=X0

iMqF

T+Op

µ1√NT

¶+Op

µ1

N

¶, (36)

X0iMXi

T=X0

iMqXi

T+Op

µ1√NT

¶+Op

µ1

N

¶, (37)

andX0

iMεiT

=X0

iMqεiT

+Op

µ1√NT

¶+Op

µ1

N

¶. (38)

where

Mq = IT − Q¡Q0Q

¢−Q0, with Q = GP. (39)

Using the above results, noting that T−1X0iMqXi = Op (1), and assuming that the rank

condition (9) is satisfied we have

bi − βi =

µX0

iMgXi

T

¶−1µX0

iMgεiT

¶+Op

µ1√NT

¶+Op

µ1

N

¶. (40)

Since εi is independently distributed ofXi andG = (D,F), then for a fixed T , limN→∞E³bi − βi

´=

0. The finite-T distribution of bi − βi will be free of nuisance parameters as N → ∞, butwill depend on the probability density of εi. For N and T sufficiently large, the distribution

of√T³bi − βi

´will be asymptotically normal if the rank condition (9) is satisfied and if N

and T are of the same order of magnitudes, namely, if T/N → κ as N and T →∞, where κis a positive finite constant. To see why this additional condition is needed, using (40) note

that√T³bi − βi

´=

µX0

iMgXi

T

¶−1X0

iMgεi√T

+Op

Ã√T

N

!+Op

µ1√N

¶, (41)

and the asymptotic distribution of√T³bi − βi

´will be free of nuisance parameters only if

√T/N → 0, as (N, T )

j→∞. For this condition to be satisfied it is sufficient that T/N → κ,

as (N,T )j→∞, where κ is a finite non-negative constant as N and T →∞.

In the case where there are no cointegrating relations amongst the elements of ft, and

d3t = 0, the results simplify since by Lemma 2

X0iMF

T=X0

iMgF

T+Op

µ1√NT

¶, (42)

10

X0iMXi

T=X0

iMgXi

T+Op

µ1√NT

¶, (43)

andX0

iMεiT

=X0

iMgεiT

+Op

µ1√NT

¶. (44)

Hence, the condition√T/N → 0, as (N,T )

j→∞ will not be needed for the validity of the

asymptotic distribution of√T³bi − βi

´for large N and T . The following theorem provides

a formal statement of these results and the associated asymptotic distributions in the case

where the rank condition (9) is satisfied.

Theorem 1 Consider the panel data model (1) and (2) and suppose that kβik < K, kΠik <K, Assumptions 1-5 hold. Let

√T/N → 0, as (N,T )

j→ ∞, and the rank condition (9) besatisfied. (a) - (N-asymptotic) The common correlated effects estimator, bi, defined by (15)

is unbiased for a fixed T > n+ 2k + 1 and N →∞, in the sense that limN→∞E³bi´= βi.

Under the additional assumption that εit ∼ IIDN(0, σ2i ),

bi − βid→ N(0,ΣT,bi), (45)

as N →∞, whereΣT,bi = T−1σ2iΨ

−1ig , Ψig = T−1 (X0

iMgXi) , (46)

Mg = IT −G(G0G)−1G0, (47)

and G = (g1,g2, ...,gT ) = (F,D). (b) - (Joint asymptotic) As (N,T )j→∞ (in no particular

order), bi is a consistent estimator of βi. Then, as (N,T )j→∞

√T³bi − βi

´d→ N(0,Σbi), (48)

where

Σbi = σ2iΣ−1i . (49)

An asymptotically unbiased estimator of ΣT,bi, as N →∞ for a fixed T > n+ 2k+ 1, is

given by:

ΣT,bi = σ2i¡X0

iMXi

¢−1, (50)

where

σ2i =

³yi −Xibi

´0M³yi −Xibi

´T − (n+m+ k)

. (51)

In the case where (N, T )j→∞, a consistent estimator of Σbi is given by

Σbi = σ2i

µX0

iMXi

T

¶−1, (52)

11

where

σ2i =

³yi −Xibi

´0M³yi −Xibi

´T − (n+ 2k + 1) . (53)

Remark 3 It is worth noting that despite the fact that under our Assumptions ft, yit and

xit are I(1) and cointegrated, in the results of Theorem 1 the rate of convergence of bi to βi

as (N,T )j→∞ is

√T and not T . It is helpful to develop some intuition behind this result.

Since for N sufficiently large ft can be well approximated by the cross section averages we

might as well consider the case where ft is observed. Abstracting from dt, and without loss

of generality substitute (2) in (1) to obtain

yit = β0i (Γ0ift + vit) + γ

0ift + εit = ϑ0ift + κit

where ϑi = Γiβi+γi and κit = εit + β0ivit. It is clear that if ft is I(1), as postulated

under our assumptions, then for all values of βi, κit is I(0) and yit and ft will be I(1) and

cointegrated. Hence, it is easily seen that ϑi can be estimated superconsistently. However, the

OLS estimator of βi need not be superconsistent. To see this note that βi can equivalently be

estimated by regressing the residuals of yit on ft on the residuals of xit on ft. Both these sets

of residuals are stationary processes and the resulting estimator of βi will be√T -consistent.

Remark 4 When the rank condition, (9), is not satisfied consistent estimation of the indi-

vidual slope coefficients is not possible.

3.1 Pooled Estimators

We now examine the asymptotic properties of the pooled estimators. Focusing first on the

MG estimator, and using (29) we have

√N³bMG − β

´=

1√N

NXi=1

υi +1

N

NXi=1

Ψ−1iT

Ã√NX0

iMF

T

!γi+

1

N

NXi=1

Ψ−1iT

Ã√NX0

iMεiT

!, (54)

In the case where the rank condition (9) is satisfied, again by Lemmas 1-3 we have√N¡X0

iMF¢

T= Op

µ1√T

¶,

and1

N

NXi=1

Ψ−1iT

Ã√NX0

iMεiT

!= Op

µ1√T

¶,

12

Thus√N³bMG − β

´=

1√N

NXi=1

υi +Op

µ1√T

¶.

Hence √N³bMG − β

´d→ N(0,ΣMG), as (N, T )

j→∞. (55)

The variance estimator for ΣMG suggested by Pesaran (2006) is given by

ΣMG =1

N − 1NXi=1

³bi − bMG

´³bi − bMG

´0, (56)

can be used. It can also be shown along identical lines to Pesaran (2006) that the mean

group estimator is valid even if the rank condition is not satisfied. The following theorem

summarises the results for the mean group estimator.

Theorem 2 Consider the panel data model (1) and (2) and suppose that Assumptions 1-5

hold. Then the Common Correlated Effects Mean Group estimator, bMG defined by (14), is

asymptotically (for a fixed T and as N →∞) unbiased for β, and as (N,T )j→∞

√N³bMG − β

´d→ N(0,ΣMG),

where ΣMG is consistently estimated by (56).

This theorem does not require that the rank condition, (9), holds for any number, m, of

unobserved factors so long as m is fixed, and does not impose any restrictions on the relative

rates of expansion of N and T . But in the case where the rank condition is satisfied As-

sumption 3 can be relaxed and the factor loadings, γi, need not follow the random coefficient

model. It would be sufficient that they are bounded.

Moving to the pooled estimator we have the following. bP defined by (17), can be written

as√N³bP − β

´=

Ã1

N

NXi=1

X0iMXi

T

!−1 "1√N

NXi=1

X0iM(Xiυi + εi)

T+ qNT

#. (57)

where

qNT =1√N

NXi=1

¡X0

iMF¢γi

T. (58)

Assuming random coefficients we note that γi = γ + ηi − η, where η = 1N

PNi=1 ηi. Hence

qNT =1√N

NXi=1

µX0

iMF

T

¶(γ − η) + 1√

N

NXi=1

µX0

iMF

T

¶ηi.

13

Recall that when the rank condition is not satisfied T−1¡X0

iMF¢= Op(1). But since X0M =

0, we haveNXi=1

X0iMF (γ − η) = NX0MF (γ − η) = 0,

and it follows that

qNT =1√N

NXi=1

µX0

iMF

T

¶ηi (59)

=1√N

NXi=1

µX0

iMqF

T

¶ηi +Op

µ1√N

¶+Op

µ1√T

¶,

Substituting this result in (57), and making use of (34) and (35) we have

√N³bP − β

´=

Ã1

N

NXi=1

X0iMqXi

T

!−1 "1√N

NXi=1

X0iMq(Xiυi + εi + Fηi)

T

#+ (60)

Op

µ1√T

¶.

Hence, as (N, T )j→∞ √

N³b− β

´d→ N(0,Σ∗P ),

where

Σ∗P = Ψ∗−1R∗Ψ∗−1, (61)

Ψ∗ = limN→∞

ÃN−1

NXi=1

Σiq

!, R∗= lim

N→∞

"N−1

NXi=1

¡ΣiqΩυΣiq +QifΩηQ

0if

¢#, (62)

and Σiq and Qif are defined by

Σiq = p limT→∞

¡T−1X0

iMqXi

¢and Qif = p lim

T→∞¡T−1X0

iMqF¢. (63)

The variance estimator for Σ∗P suggested by Pesaran (2006) and given by

Σ∗P = Ψ∗−1R∗Ψ∗−1, (64)

where

Ψ∗ = N−1NXi=1

µX0

iMXi

T

¶, (65)

R∗ =1

(N − 1)NXi=1

µX0

iMXi

T

¶³bi − bMG

´³bi − bMG

´0µX0iMXi

T

¶. (66)

Again we summarise these results in the following theorem.

14

Theorem 3 Consider the panel data model (1) and (2) and suppose that Assumptions 1-5

hold. Then the Common Correlated Effects Pooled estimator, bP , defined by (17) is asymp-

totically unbiased for β, and as (N, T )j→∞ we have

√N³bP − β

´d→ N(0,Σ∗P ),

where Σ∗P is given by (61)-(63).

Overall we see that despite a number of differences in the above analysis, especially

in terms of the results given in (33)-(35), compared to the results in Pesaran (2006), the

conclusions are remarkably similar when the factors are assumed to follow unit root processes

rather that weakly stationary ones.

4 Monte Carlo Design and Evidence

In this section we provide Monte Carlo evidence on the small sample properties of the

CCEMG and the CCEP estimators. We also consider the two alternative principal compo-

nent augmentation approaches discussed in Kapetanios and Pesaran (2006). The first PC

approach applies the Bai and Ng (2002) procedure to zit = (yit,x0it )0 to obtain consistent

estimates of the unobserved factors, and then uses the estimated factors to augment the

regression (1), and thus produces consistent estimates of β. We consider both pooled and

mean group versions of this estimator which we refer to as PC1POOL and PC1MG. The

second PC approach consists of extracting the principal component estimates of the unob-

served factors from yit and xit separately, regressing yit and xit on their respective factor

estimates separately, and then applying the standard pooled and mean group estimators,

with no cross-sectional dependence adjustments, to the residuals of these regressions. We

refer to the estimators based on this approach as PC2POOL and PC2MG, respectively.

The experimental design of the Monte Carlo study is closely related to the one used in

Pesaran (2006). Consider the following data generating process (DGP):

yit = αi1d1t + βi1x1it + βi2x2it + γi1f1t + γi2f2t + εit, (67)

and

xijt = aij1d1t + aij2d2t + γij1f1t + γij3f3t + vijt, j = 1, 2, (68)

for i = 1, 2, ..., N , and t = 1, 2, ..., T . This DGP is a restricted version of the general linear

model considered in Pesaran (2006), and sets n = k = 2, and m = 3, with α0i = (αi1, 0),

β0i = (βi1, βi2), and γ0i = (γi1, γi2, 0), and

A0i =

µai11 ai12ai21 ai22

¶, Γ0i =

µγi11 0 γi13γi21 0 γi23

¶.

15

The observed common factors and the individual specific errors of xit are generated as

independent stationary AR(1) processes with zero means and unit variances:

d1t = 1, d2t = ρdd2,t−1 + vdt, t = −49, ...1, ..., T ,vdt ∼ IIDN(0, 1− ρ2d), ρd = 0.5, d2,−50 = 0,

vijt = ρvijvijt−1 + υijt, t = −49, ...1, ..., T,υijt ∼ IIDN

¡0, 1− ρ2vij

¢, vji,−50 = 0,

and

ρvij ∼ IIDU [0.05, 0.95] , for j = 1, 2,

but the unobserved common factors are generated as non-stationary processes:

fjt = fjt−1 + vfj,t, for j = 1, 2, 3, t = −49, .., 0, .., T,vfj,t ∼ IIDN(0, 1), fj,−50 = 0, for j = 1, 2, 3.

The first 50 observations are discarded.

To illustrate the robustness of the CCE and PC estimators to the dynamics of the indi-

vidual specific errors of yit, these are generated as the (cross sectional) mixture of stationary

heterogeneous AR(1) and MA(1) errors. Namely,

εit = ρiεεi,t−1 + σi

q1− ρ2iεωit, i = 1, 2, ..., N1, t = −49, .., 0, .., T,

and

εit =σip1 + θ2iε

(ωit + θiεωi,t−1) , i = N1 + 1, ..., N , t = −49, .., 0, .., T,

where N1 is the nearest integer of N/2,

ωit ∼ IIDN (0, 1) , σ2i ∼ IIDU [0.5, 1.5] , ρiε ∼ IIDU [0.05, 0.95] , θiε ∼ IIDU [0, 1] .

ρvij, ρiε, θiε and σi are not changed across replications. The first 49 observations are dis-

carded. The factor loadings of the observed common effects, αi1, and vec(Ai) = (ai11, ai21, ai12, ai22)0

are generated as IIDN(1, 1), and IIDN(0.5τ4, 0.5 I4), where τ4 = (1, 1, 1, 1)0, and are not

changed across replications. They are treated as fixed effects. The parameters of the unob-

served common effects in the xit equation are generated independently across replications

as

Γ0i =µ

γi11 0 γi13γi21 0 γi23

¶∼ IID

µN (0.5, 0.50) 0 N (0, 0.50)N (0, 0.50) 0 N (0.5, 0.50)

¶.

16

For the parameters of the unobserved common effects in the yit equation, γi, we considered

two different sets that we denote by A and B. Under set A, γi are drawn such that the rank

condition is satisfied, namely

γi1 ∼ IIDN (1, 0.2) , γi2A ∼ IIDN (1, 0.2) , γi3 = 0,

and

E³ΓiA

´= (E (γiA) , E (Γi)) =

⎛⎝ 1 0.5 01 0 00 0 0.5

⎞⎠ .

Under set Bγi1 ∼ IIDN (1, 0.2) , γi2B ∼ IIDN (0, 1) , γi3 = 0,

so that

E³ΓiB´= (E (γiB) , E (Γi)) =

⎛⎝ 1 0.5 00 0 00 0 0.5

⎞⎠ ,

and the rank condition is not satisfied. For each set we conducted two different experiments:

• Experiment 1 examines the case of heterogeneous slopes with βij = 1+ ηij, j = 1, 2,

and ηij ∼ IIDN(0, 0.04), across replications.

• Experiment 2 considers the case of homogeneous slopes with βi = β = (1, 1)0.

The two versions of experiment 1 will be denoted by 1A and 1B, and those of experiment2 by 2A and 2B. For this Monte Carlo study we also computed the CCEMG and the CCEPestimators as well as the associated “infeasible” estimators (MG and Pooled) that include

f1t and f2t in the regressions of yit on (d1t,xit), and the “naive” estimators that excludes

these factors. The naive estimators illustrate the extent of bias and size distortions that can

occur if the error cross section dependence is ignored.

In relation to the infeasible pooled estimator, it is important to note that this estimator

although unbiased under all the four sets of experiments, it need not be efficient since in

these experiments the slope coefficients, βi, and/or error variances, σ2i , differ across i. As a

result the CCE or PC augmented estimators may in fact dominate the infeasible estimator

in terms of RMSE, particularly in the case of experiments 1A and 1B where the slopes aswell as the error variances are allowed to vary across i.

Another important consideration worth bearing in mind when comparing the CCE and

the PC type estimators is the fact that the computation of the PC augmented estimators

assumes thatm = 3, the number of unobserved factors, is known. In practice, m might

be difficult to estimate accurately particularly when N or T happen to be smaller than 50.

17

By contrast the CCE type estimators are valid for any fixed m and do not require a prior

estimate for m.

Each experiment was replicated 2000 times for the (N, T ) pairs withN, T = 20, 30, 50, 100, 200.

In what follows we shall focus on β1 (the cross section mean of βi1). Results for β2 are very

similar and will not be reported. Finally, for completeness we state below the exact formulae

for the variance estimators used for the different estimators. The non-parametric variance

estimators of the mean group estimators, bMG = N−1PNi=1 bi, are computed as

dV ar(bMG) =1

N (N − 1)NXi=1

³bi − bMG

´³bi − bMG

´0, (69)

where

bi =³X0

iMxXi

´−1X0

iMxMyyi,

Mx = IT − Hx

³H0

xHx

´−H0

x, My = IT − Hy

³H0

yHy

´−H0

y.

For the CCEMG estimator, Hx= Hy = H = (D, Z), so that bi = bi, which is defined

by (15); for the PC1MG estimator, Hx= Hy = Fz, where Fz is a T × (n + m) matrix of

extracted factors from Zi for all i, together with observed common factors; for the PC2MG

estimator Hx = Fx and Hy = Fy, where Fx and Fy are T × (nx +mx) and T × (ny +my)

matrices of extracted factors fromXi and yi respectively for all i, together with the observed

common factors with nx and ny being the number of observed common factors in Xi and

yi respectively, and mx and my defined similarly; for the infeasible mean group estimator,

Hx= Hy = Fy, which is a T ×my matrix of unobserved factors in yi; for the naive mean

group estimator, Hx= Hy = D. Next, the non-parametric variance of the pooled estimator,

bP , is computed as dV ar(bP ) = N−1Ψ−1RΨ−1, (70)

where

bP =

ÃNXi=1

X0iMxXi

!−1 NXi=1

X0iMxMyyi,

Ψ = N−1NXi=1

ÃX0

iMxXi

T

!,

R =1

(N − 1)NXi=1

ÃX0

iMxXi

T

!³bi − bMG

´³bi − bMG

´0ÃX0iMxXi

T

!.

4.1 Results

Results of experiments 1A, 2A, 1B, 2B are summarized in Tables 1 to 4, respectively. Wealso provide results for the naive estimator (that excludes the unobserved factors or their

18

estimates) and the infeasible estimator (that includes the unobserved factors as additional

regressors) for comparison purposes. But for the sake of brevity we include the simulation

results for these estimators only for experiment 1A reported in Table 1.As can be seen from Table 1 the naive estimator is substantially biased, performs very

poorly and is subject to large size distortions; an outcome that continues to apply in the

case of other experiments (not reported here). In contrast, the feasible CCE estimators

perform well, have bias that are close to the bias of the infeasible estimators, show little size

distortions even for relatively small values of N and T , and their RMSE falls steadily with

increases in N and/or T . These results are quite similar to the results presented in Pesaran

(2006), and illustrate the robustness of the CCE estimators to the presence of unit roots in

the unobserved common factors. This is important since it obviates the need for pre-testing

involving unobserved factors.

The CCE estimators perform well, in both heterogeneous and homogeneous cases, and

irrespective of whether the rank condition is satisfied, although the CCE estimators with

rank deficiency have sightly higher RMSEs than those with full rank. The RMSEs of the

CCE estimators of Tables 1 and 3 (heterogeneous case) are higher than those reported in

Tables 2 and 4 for the homogeneous case. The sizes of the t-test based on the CCE estimators

are very close to the nominal 5% level. In the case of full rank, the power of the tests for the

CCE estimators are much higher than in the rank deficient case. Finally, not surprisingly

the power of the tests for the CCE estimators in the homogeneous case is higher than that

in the heterogeneous case.

It is also important to note that the small sample properties of the CCE estimator does

not seem to be much affected by the residual serial correlation of the idiosyncratic errors,

εit. The robustness of the CCE estimator to the short run dynamics is particularly helpful

in practice where typically little is known about such dynamics. In fact a comparison of the

results for the CCEP estimator with the infeasible counterpart given in Table 1 shows that

the former can even be more efficient (in the RMSE sense). For example the RMSE of the

CCEP for N = T = 50 is 3.97 whilst the RMSE of the infeasible pooled estimator is 4.31.

This might seem counter intuitive at first, but as indicated above the infeasible estimator

does not take account of the residual serial correlation of the idiosyncratic errors, but the

CCE estimator does allow for such possibilities indirectly through the use of the cross section

averages that partly embody the serial correlation properties of ft and εit’s.

Consider now the PC augmented estimators and recall that they are computed assuming

the true number of common factors is known. The results summarized in Tables 1-4 bear

some resemblance to those presented in Kapetanios and Pesaran (2006). The bias and

RMSEs of the PC1POOL and PC1MG estimators improve as both N and T increase, but

19

the t-tests based on these estimators substantially over-reject the null hypothesis. The

PC2POOL and PC2MG estimators perform even worse. The biases of the PC estimators

are always larger in absolute value than the respective biases of the CCE estimators. The

size distortion of the PC augmented estimators is particularly pronounced in the case of

the experiments 1A and 2A (in Tables 1 and 2) where the full rank conditions are met. It

is also interesting that in the case of some of the experiments the bias distortions of the

tests based on the PC augmented estimators do not improve even for relatively large N

and T . An interesting distinction arises when comparing results for experiments 1A and

1B. For 1A (heterogeneous slopes and full rank) results are very poor for small values of Nand T but improve considerably as N rises and less perceptibly as T rises. For experiment

1B (heterogeneous slopes and rank deficient) results are much better for small values of Nand T . Finally, it is worth noting that in both cases the performance of the PC estimators

actually get worse when N is small and kept small but T rises. This may be related to the

fact that the accuracy of the factor estimates depends on the minimum of N and T .

5 Conclusions

Recently, there has been increased focus in the panel data literature on problems arising in

estimation and inference when the standard assumption that the errors of the panel regression

are cross-sectionally uncorrelated, is violated. When the errors of a panel regression are

cross-sectionally correlated then standard estimation methods do not necessarily produce

consistent estimates of the parameters of interest. An influential strand of the relevant

literature provides a convenient parametrisation of the problem in terms of a factor model

for the error terms.

Pesaran (2006) adopts an error multifactor structure and suggests new estimators that

take into account cross-sectional dependence, making use of cross-sectional averages of the

dependent and explanatory variables. However, he focusses on the case of weakly stationary

factors that could be restrictive in some applications. This paper provides a formal extension

of the results of Pesaran (2006) to the case where the unobserved factors are allowed to follow

unit root processes. It is shown that the main results of Pesaran continue to hold in this

more general case. This is certainly of interest given the fact that usually there is a large

difference between results obtained for unit root and stationary processes. When we consider

the small sample properties of the new estimators, we observe that again the results accord

with the conclusions reached in the stationary case, lending further support to the use of the

CCE estimators irrespective of the order of integration of the data observed.

20

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22

6 Appendix

Lemma 1 Under Assumptions 1-5 and ifP∞

=0W is of reduced rank whereW , = 0, ...,

is defined in Remark 2, then as (N, T )j→∞,

X0iMXi

T=X0

iMqXi

T+Op

µ1√NT

¶+Op

µ1

N

¶, (71)

X0iMF

T=X0

iMqF

T+Op

µ1√NT

¶+Op

µ1

N

¶, (72)

X0iMεiT

=X0

iMqεiT

+Op

µ1√NT

¶+Op

µ1

N

¶. (73)

Proof. We start by proving (71). The proof for (72) and (73) follow similarly. We

need to determine the order of probability of X0iMXi

T− X0

iMqXi

T. We split the problem in two

parts. By remark 2, ft may be represented, via a common trends representation, by a set of

r stationary components and m − r random walk components. Without loss of generality

we disregard the deterministic component d1t in the analysis. Let K1 denote the matrix

of observations on the nonstationary components of the common trends representation of

ft and the nonstationary components of dt. Let K2 denote the matrix of observations on

the stationary components of the common trends representation of ft and the stationary

components of dt. Note that the transformations needed to get (K1,K2) from G simply

involve nonsingular rotations of ft. Let

Hj = KjPj + U∗,

Mjq = IT − Qj

¡Q0

jQj

¢−Q0

j,

Mj = IT − Hj

¡H0

jHj

¢−Hj,

with Qj = KjPj for j = 1, 2 where Pj is a nonsingular transformation of a partition of P

comformable to (K1,K2). Then, the order of probability ofX0iMXi

T− X0

iMqXi

Tis equal to the

maximum of the orders of probability of X0iM1Xi

T− X0

iM1qXi

Tand X0

iM2Xi

T− X0

iM2qXi

Twhere Xi

are the residuals of Xi when regressed on K1. To see this note that

X0iMXi

T=X0

iM2Xi

T,

andX0

iMqXi

T=X0

qiM2qXqi

T,

where Xi = M1Xi, Xqi = M1qXi,

M2 = IT − M1H2

¡H02M1H2

¢−H02M1,

23

and

M2q = IT − M1qQ2

¡Q02M1qQ2

¢−Q02M1q.

Then

X0iMXi

T− X

0iMqXi

T≤ C

⎛⎝°°°°°X0i

¡M1 − M1q

¢M2Xi

T

°°°°°+°°°°°°X0

i

³M2 − M2q

´Xi

T

°°°°°°⎞⎠ ,

for some positive constant, C. The desired result then follows easily from Lemma 2, Lemmas

A.2 and A.3 of Pesaran (2006).

Lemma 2 Under Assumptions 1-5 and ifP∞

=0W is of full rank whereW , = 0, ..., is

defined in Remark 2, then as (N, T )j→∞,

X0iMXi

T=X0

iMqXi

T+Op

µ1√NT

¶, (74)

X0iMF

T=X0

iMqF

T+Op

µ1√NT

¶, (75)

X0iMεiT

=X0

iMqεiT

+Op

µ1√NT

¶. (76)

Proof. For simplicity but without loss of generality we assume that d3t is empty. We

start by proving (74). Throughout the proof Ci, i = 1, ..., denote different positive constants.

We need to determine the order of probability of X0iMXi

T− X0

iMqXi

T. But this is equal to

X0iH¡H0H

¢−H0Xi

T− X

0iQ¡Q0Q

¢−Q0Xi

T=

µX0

iH

T 3/2

¶µH0HT 2

¶−µH0Xi

T 3/2

¶−µX0

iQ

T 3/2

¶µQ0QT 2

¶−µQ0Xi

T 3/2

¶≤

C1

°°°°X0iH

T 3/2− X

0iQ

T 3/2

°°°°+ C2

°°°°H0HT 2− Q

0QT 2

°°°° . (77)

We examine the first term of (77) first. We have

X0iH

T 3/2− X

0iQ

T 3/2=X0

iU

T 3/2,

where U = 1N

PNi=1Ui. But

X0iU

T 3/2≤ C3

°°°°F0UT 3/2+D0UT 3/2

+V0

iU

T 3/2

°°°° , (78)

andV0

iU

T 3/2= Op

µ1

T√N

¶+Op

µ1

N√T

¶, (79)

24

by Lemma A.2 of Pesaran (2006). A similar treatment can be applied to the first and second

terms of the RHS of (78). We therefore examine the first term only. Consider the th row

of T−3/2¡F0U

¢and note that it can be written as T−3/2

³PTt=1 f tu

0t

´. Consider now the

limit of T−3/2PT

t=1 f tut. First note that Assumptions 1-3 and Remark 1, via Theorem 4.1

of Davidson and De Jong (2000), guarantee that uit can play the role of the integrator in

a functional central limit theorem, i.e. the limit of T−1PT

t=1 f tu0it is a stochastic integral

and therefore Op (1). Further note that for all finite N as well as when N →∞, the limit ofT−1

PTt=1 f t

√N ut is a stochastic integral since assumption 3 and Remark 1 state that u0it

is a L2−NED process of size 1/2. This assumption, in turn, implies that ut is a L2−NED

process of size 1/2 for all finite N and as N →∞, and the result holds again via Theorem4.1 of Davidson and De Jong (2000). To see this we need to show that for some positive

integer s > 0

E

(1√N

NXi=1

uit −E

Ã1√N

NXi=1

uit

¯¯ t− s

!)2= O(s−1). (80)

But

E

(1√N

NXi=1

uit −E

Ã1√N

NXi=1

uit

¯¯ t− s

!)2≤ 1

N

NXi=1

E [uit − E (uit|t− s)]2 , (81)

and since E [uit −E (uit|t− s)]2 = O(s−1) for all i, the desired result follows. From the

above we also haveF0UT 3/2

= Op

µ1√NT

¶. (82)

Similarly,D0UT 3/2

= Op

µ1√NT

¶, (83)

and hence,X0

iU

T 3/2= Op

µ1√NT

¶. (84)

We next examine the second term of (77). We have

H0HT 2− Q

0QT 2≤ C4

°°°°Q0UT 2

+U0UT 2

°°°° . (85)

By Lemma A.2 of Pesaran (2006) it follows that

U0UT 2

= Op

µ1

NT

¶. (86)

Further,Q0UT 2

= Op

µF0UT 2

¶+Op

µD0UT 2

¶= Op

µ1

T√N

¶, (87)

25

by (82). Thus,H0HT 2− Q

0QT 2

= Op

µ1

T√N

¶. (88)

We now consider (75). We have

X0iH¡H0H

¢−H0F

T− X

0iQ¡Q0Q

¢−Q0F

T=

µX0

iH

T 3/2

¶µH0HT 2

¶−µH0FT 3/2

¶−µX0

iQ

T 3/2

¶µQ0QT 2

¶−µQ0FT 3/2

¶≤

C5

°°°°X0iH

T 3/2− X

0iQ

T 3/2

°°°°+ C6

°°°°H0HT 2− Q

0QT 2

°°°°+ C7

°°°°H0FT 3/2

− Q0F

T 3/2

°°°° . (89)

We only need to examine the third term of the RHS of (89). But

H0FT 3/2

− Q0F

T 3/2=U0FT 3/2

.

But then by (82) and (84), (75) follows.

Finally, we consider (76). The treatment follows closely that in (77) and (89). In partic-

ularX0

iH¡H0H

¢−H0εi

T− X

0iQ¡Q0Q

¢−Q0εi

T=µ

X0iH

T 3/2

¶µH0HT 2

¶−µH0εiT 3/2

¶−µX0

iQ

T 3/2

¶µQ0QT 2

¶−µQ0εiT 3/2

¶≤

C8

°°°°X0iH

T 3/2− X

0iQ

T 3/2

°°°°+ C9

°°°°H0HT 2− Q

0QT 2

°°°°+ C10

°°°°H0εiT 3/2

− Q0εi

T 3/2

°°°° . (90)

But,H0εiT 3/2

− Q0εi

T 3/2=U0εiT 3/2

.

By Lemma A.2 of Pesaran (2006), it follows that

U0εiT 3/2

= Op

µ1

T√N

¶+Op

µ1

N√T

¶.

Hence, (76) follows.

26

Table 1: Small Sample Properties of Common Correlated Effects Type Estimatorsin the Case of Experiment 1A (Heterogeneous Slopes + Full Rank)

Bias (×100) Root Mean Square Errors (×100) Size (5% level, H0 : β1 = 1.00) Power (5% level, H1 : β1 = 0.95)CCE Type Estimators(N,T) 20 30 50 100 200 20 30 50 100 200 20 30 50 100 200 20 30 50 100 200

CCEMG20 0.05 -0.10 -0.03 0.06 -0.07 9.67 7.89 6.74 5.87 5.54 7.20 6.90 7.15 7.90 7.55 11.65 13.00 16.10 17.50 20.1030 0.09 -0.01 -0.01 -0.13 0.10 7.69 6.09 5.11 4.54 4.22 6.95 5.30 5.90 6.25 6.35 11.40 14.25 18.05 22.05 26.8550 -0.19 0.22 -0.11 0.14 -0.04 5.88 4.61 4.01 3.44 3.13 5.70 5.05 6.65 6.20 5.95 15.10 20.40 25.60 34.10 36.65100 0.00 0.04 0.04 0.03 0.04 4.25 3.46 2.89 2.33 2.27 5.75 5.85 5.25 4.90 6.20 23.35 34.30 44.40 56.00 63.25200 -0.05 -0.02 -0.03 0.05 0.00 3.07 2.49 2.01 1.72 1.51 4.40 5.15 4.90 5.60 5.10 35.55 52.65 68.70 83.65 90.50

CCEP20 0.18 0.00 -0.05 -0.01 -0.13 8.75 7.67 6.85 6.32 6.21 7.70 8.10 7.30 8.05 7.15 12.75 13.50 16.05 16.80 18.3030 -0.17 -0.12 0.09 -0.15 0.13 7.10 5.99 5.32 4.78 4.46 7.55 6.25 6.75 6.65 6.45 12.40 15.00 19.30 20.65 26.9050 0.00 0.18 -0.07 0.12 -0.01 5.33 4.51 3.97 3.47 3.22 6.80 6.20 5.90 6.35 6.45 17.45 22.15 26.40 32.90 36.25100 0.00 0.09 0.03 0.00 0.02 3.78 3.25 2.85 2.34 2.28 5.70 5.65 5.60 5.15 6.25 28.15 37.40 44.80 55.20 61.75200 -0.07 -0.04 -0.05 0.05 0.00 2.71 2.29 1.95 1.70 1.53 5.10 4.35 5.05 4.70 4.75 44.75 56.80 70.30 83.55 89.75

Principal Component Estimators, AugmentedPC1MG

20 -12.27 -11.15 -10.30 -8.87 -8.90 17.09 14.81 13.24 11.51 11.55 22.55 25.35 30.05 33.40 37.40 12.15 12.95 13.30 12.70 13.7530 -9.25 -7.86 -6.46 -5.72 -5.25 13.55 10.84 8.98 7.80 7.15 20.60 20.90 21.65 24.75 24.70 10.75 8.25 7.35 7.40 6.7550 -6.84 -5.05 -3.89 -3.01 -3.12 10.10 7.79 5.86 4.67 4.47 19.95 17.65 16.25 14.95 17.90 8.70 8.20 7.65 11.40 9.75100 -4.78 -3.21 -2.03 -1.57 -1.45 7.44 5.34 3.68 2.87 2.72 20.10 16.80 11.45 9.75 11.10 9.55 12.15 20.25 28.85 36.75200 -4.31 -2.54 -1.39 -0.81 -0.78 6.39 4.19 2.60 1.93 1.71 25.20 17.95 10.95 8.15 7.65 13.85 21.95 42.85 67.65 77.15

PC1POOL20 -11.97 -11.04 -10.35 -9.09 -9.23 15.88 14.38 13.07 11.59 12.07 25.50 28.35 32.05 34.45 38.95 12.05 14.10 14.90 14.55 14.9030 -8.86 -7.66 -6.34 -5.73 -5.37 12.48 10.45 8.89 7.80 7.34 21.45 23.75 22.05 24.70 25.50 11.00 8.80 7.55 7.95 6.3550 -6.20 -4.86 -3.81 -3.07 -3.19 9.06 7.52 5.72 4.73 4.54 21.40 18.75 16.00 16.05 18.90 8.55 9.55 8.10 10.90 9.65100 -4.36 -3.00 -2.01 -1.60 -1.49 6.61 5.01 3.61 2.88 2.74 21.05 16.85 11.25 9.35 10.80 11.25 14.55 20.85 27.90 36.30200 -3.62 -2.32 -1.36 -0.81 -0.79 5.39 3.81 2.51 1.91 1.73 25.15 17.60 10.50 7.80 7.80 16.35 26.75 45.45 68.00 76.15

Principal Component Estimators, OrthogonalisedPC2MG

20 -31.26 -27.06 -24.01 -22.67 -23.11 32.83 28.34 25.00 23.44 23.83 86.50 88.45 91.25 95.20 97.40 74.10 73.95 75.80 82.05 88.2030 -25.50 -21.21 -18.27 -16.69 -16.33 26.82 22.25 19.13 17.35 16.92 86.85 87.10 89.10 93.35 95.95 70.15 67.80 66.10 69.25 74.7050 -20.65 -16.23 -13.32 -11.41 -10.89 21.68 17.06 13.98 11.95 11.37 90.15 88.35 88.80 89.05 91.70 70.80 60.25 52.20 45.80 46.10100 -16.17 -12.44 -9.69 -7.61 -6.60 16.87 12.97 10.18 7.99 7.02 93.65 93.30 89.75 87.50 83.30 72.35 56.20 37.60 19.30 13.60200 -14.61 -10.78 -8.12 -5.79 -4.59 15.11 11.19 8.45 6.08 4.85 98.95 97.85 95.45 90.75 83.75 79.65 60.20 33.30 10.00 6.75

PC2POOL20 -31.97 -27.47 -24.27 -23.18 -24.19 33.39 28.69 25.23 23.99 24.99 91.00 90.70 93.20 95.55 98.50 80.65 78.60 78.80 83.35 90.4530 -26.32 -21.51 -18.24 -16.83 -16.75 27.53 22.48 19.13 17.51 17.37 91.35 90.40 89.70 93.35 96.15 78.50 71.80 66.65 70.65 76.9050 -21.22 -16.35 -13.17 -11.35 -10.99 22.10 17.15 13.82 11.91 11.48 95.05 90.90 88.95 88.20 91.70 79.65 63.80 52.95 46.20 48.25100 -16.77 -12.52 -9.62 -7.55 -6.60 17.43 13.06 10.11 7.95 7.03 97.95 95.05 90.50 86.45 82.30 80.90 60.80 38.10 18.30 14.25200 -15.16 -10.91 -8.00 -5.66 -4.53 15.67 11.33 8.34 5.96 4.79 99.75 98.45 95.95 89.35 82.50 88.65 65.85 33.35 8.40 6.30

(Table 1 Continued)Bias (×100) Root Mean Square Errors (×100) Size (5% level, H0 : β1 = 1.00) Power (5% level, H1 : β1 = 0.95)

CCE Type EstimatorsInfeasible Estimators (including f1t and f2t)(N,T) 20 30 50 100 200 20 30 50 100 200 20 30 50 100 200 20 30 50 100 200

Infeasible MG20 0.01 -0.19 -0.08 0.15 -0.08 7.21 6.33 5.62 4.98 4.76 6.40 6.20 6.80 5.95 6.50 12.75 15.35 16.85 19.70 20.4030 0.02 -0.14 0.01 -0.02 0.12 5.91 4.95 4.43 3.97 3.87 6.50 5.80 6.05 5.30 5.90 16.15 18.05 23.35 25.20 28.8050 -0.10 0.07 -0.06 0.14 -0.04 4.48 3.75 3.39 3.09 2.94 6.45 5.25 5.90 5.25 5.20 21.70 27.35 31.45 38.45 40.25100 0.01 0.07 0.02 0.00 0.04 3.16 2.78 2.49 2.15 2.14 5.50 5.15 5.45 4.70 5.45 36.85 46.15 55.10 62.50 66.65200 -0.07 0.04 -0.07 0.06 0.01 2.22 1.93 1.69 1.57 1.44 4.85 5.00 5.00 5.60 4.70 59.15 72.85 82.25 90.40 92.75

Infeasible Pooled20 0.15 -0.13 -0.15 -0.26 -0.21 7.30 6.96 6.92 7.11 7.40 6.40 6.80 6.60 7.00 5.10 13.70 13.75 14.55 14.10 12.6530 -0.20 -0.15 0.22 -0.07 0.27 6.23 5.78 5.79 5.89 6.61 7.05 5.90 7.00 5.25 5.70 15.70 15.35 18.95 16.70 16.6050 0.12 0.07 -0.08 0.21 0.02 4.61 4.40 4.31 4.71 5.02 5.70 5.80 5.50 6.25 5.00 22.20 22.55 23.65 25.50 21.00100 -0.05 0.07 0.09 0.06 0.00 3.30 3.26 3.12 3.30 3.52 5.25 5.60 5.20 5.20 5.30 33.45 38.20 38.85 36.75 32.30200 -0.08 0.06 -0.12 0.07 -0.02 2.35 2.22 2.20 2.45 2.49 4.95 4.70 4.50 5.85 4.70 56.15 62.10 59.50 59.05 52.20

Naïve Estimators (excluding f1t and f2t)Naïve MG

20 22.18 23.13 26.82 29.96 32.62 31.76 32.97 37.37 41.49 47.04 32.05 32.95 34.85 35.45 31.50 41.00 42.65 43.50 41.95 38.0530 22.23 25.06 28.36 31.33 34.01 30.51 33.31 37.87 41.46 45.32 40.45 44.10 46.65 43.85 39.45 51.00 53.95 57.45 52.20 47.1550 22.21 23.91 25.65 29.61 33.64 29.75 31.12 32.75 37.73 42.66 55.80 59.30 58.00 59.25 54.75 68.30 70.85 70.30 69.20 65.05100 21.97 23.92 26.76 30.04 32.88 28.40 30.02 32.97 36.39 40.06 71.20 75.25 77.90 78.60 75.25 81.05 84.35 85.95 85.85 83.20200 22.15 24.09 27.49 30.09 33.23 27.87 29.44 32.80 35.71 39.34 81.85 86.00 87.85 88.05 87.95 88.75 91.95 92.30 92.90 92.05

Naïve Pooled20 25.25 26.60 31.27 33.59 34.84 35.30 37.01 42.66 45.42 47.67 42.15 43.65 47.75 45.20 44.50 52.50 52.65 55.95 53.40 51.9530 25.76 29.39 32.45 35.37 35.46 35.48 39.13 42.70 45.97 46.81 51.55 56.70 57.65 59.55 56.20 61.05 66.60 66.55 67.75 64.5550 26.54 28.75 30.39 34.01 35.88 35.61 37.39 39.05 44.04 45.93 64.75 67.15 69.25 70.35 69.35 73.55 76.25 78.25 78.65 77.45100 25.81 28.47 31.30 33.15 34.91 34.39 36.76 39.90 41.79 44.27 75.85 78.90 81.35 79.30 80.15 85.10 86.55 88.05 86.65 86.40200 25.95 28.32 31.89 33.65 34.11 34.20 36.21 39.63 42.39 42.68 83.45 86.25 87.70 87.40 87.20 89.95 91.90 93.55 92.20 92.20

Notes: The DGP is yit = αi1d1t + βi1x1it + βi2x2it + γi1f1t + γi2f2t + εit with εit = ρiεεi,t−1 + σi(1 − ρ2iε)1/2ωit, i = 1, 2, ..., [N/2], and εit = σi(1 + θ2iε)

−1/2 (ωit + θiεωi,t−1),i = [N/2] + 1, ..., N , ωit ∼ IIDN (0, 1), σ2i ∼ IIDU [0.5, 1.5], ρiε ∼ IIDU [0.05, 0.95], θiε ∼ IIDU [0, 1]. Regressors are generated by xijt = aij1d1t + aij2d2t + γij1f1t + γij3f3t+vijt,j = 1, 2, for i = 1, 2, ..., N . d1t = 1, d2t = 0.5d2,t−1 + vdt, vdt ∼ IIDN(0, 1− 0.52), d2,−50 = 0; fjt = fjt−1 + vfj,t, vfj,t ∼ IIDN(0, 1), fj,−50 = 0, for j = 1, 2, 3; vijt = ρvijvijt−1 + υijt,υijt ∼ IIDN(0, 1 − ρ2vij), vij,−50 = 0 and ρvij ∼ IIDU [0.05, 0.95] for j = 1, 2, for t = −49, ..., T with the first 50 observations discarded; αi1 ∼ IIDN (1, 1); aij ∼ IIDN (0.5, 0.5) for

j = 1, 2, = 1, 2; γi11 and γi23 ∼ IIDN (0.5, 0.50), γi13 and γi21 ∼ IIDN (0, 0.50); γi1 and γi2 ∼ IIDN (1, 0.2); βij = 1 + ηij with ηij ∼ IIDN(0, 0.04) for j = 1, 2. ρvij , ρiε, θiε, σ2i ,

αi1, aij for j = 1, 2, = 1, 2 are fixed across replications. CCEMG and CCEP are defined by (14) and (17), and their variance estimators are defined by (56) and (64), respectively. Thevariance estimators of all other mean group and pooled estimators are defined by (69) and (70), respectively. The PC type estimators are computed assuming the number of unobservedfactors, m = 3, is known. All experiments are based on 2000 replications.

Table 2: Small Sample Properties of Common Correlated Effects Type Estimatorsin the Case of Experiment 2A (Homogeneous Slopes + Full Rank)


CCEMG20 0.05 -0.15 0.02 -0.15 0.09 8.45 6.29 5.10 3.78 3.14 7.15 6.40 6.80 6.75 6.85 11.70 13.80 21.75 31.25 47.9030 -0.14 0.12 0.04 0.03 0.00 6.44 5.11 3.80 2.67 2.07 6.05 6.75 7.25 6.40 6.45 12.70 20.45 30.70 50.90 71.6050 0.08 -0.06 0.02 0.05 0.03 5.08 3.79 2.80 1.94 1.39 6.10 5.90 4.85 5.40 5.35 18.00 26.90 44.45 75.65 95.00100 -0.04 -0.08 0.06 -0.04 -0.01 3.59 2.76 2.02 1.35 0.98 4.55 5.50 6.05 5.10 6.10 28.30 43.00 72.35 95.20 99.90200 0.06 -0.02 0.03 0.01 0.00 2.83 2.05 1.52 1.00 0.68 5.60 4.45 6.35 5.20 5.70 44.20 67.95 91.90 99.90 100.00

CCEP20 0.18 0.00 0.03 -0.14 0.08 6.95 5.56 4.94 3.98 3.74 6.60 6.75 7.30 6.75 6.80 14.25 16.25 25.25 33.70 46.2530 -0.14 0.14 0.07 0.01 0.01 5.20 4.50 3.55 2.67 2.26 5.10 5.90 7.25 6.25 6.40 15.25 24.55 34.90 52.95 70.7050 0.05 0.07 -0.02 0.04 0.03 4.08 3.29 2.56 1.84 1.39 5.40 5.40 5.45 6.20 5.30 24.60 34.35 51.70 78.65 95.00100 -0.02 -0.04 0.06 -0.04 -0.01 2.87 2.37 1.78 1.24 0.93 5.60 6.20 6.40 5.25 5.95 41.65 58.35 81.85 97.80 100.00200 0.07 -0.03 0.01 0.02 0.00 2.17 1.63 1.32 0.92 0.65 5.60 3.95 5.70 5.60 5.35 65.25 84.40 96.95 100.00 100.00


20 -12.34 -11.39 -10.06 -9.44 -8.84 16.76 14.48 12.18 11.24 10.92 24.55 32.55 39.50 58.50 71.95 13.05 15.30 13.80 19.00 21.2030 -9.35 -7.83 -6.39 -5.66 -5.34 12.96 10.55 8.18 6.93 6.15 22.10 26.10 32.55 46.60 68.25 9.55 10.60 8.40 10.25 10.1050 -7.05 -5.28 -3.81 -3.08 -3.17 10.12 7.38 5.11 3.86 3.73 23.55 24.05 25.30 35.65 56.85 10.40 9.10 9.45 20.80 30.00100 -5.00 -3.45 -2.04 -1.64 -1.57 7.19 5.16 3.14 2.20 1.90 22.60 22.00 16.50 22.45 35.70 9.90 14.45 31.05 64.90 91.50200 -4.23 -2.65 -1.27 -0.87 -0.79 6.27 4.11 2.13 1.37 1.06 28.05 22.90 14.90 16.85 21.95 16.15 30.55 67.35 97.55 100.00

PC1POOL20 -11.78 -11.12 -9.89 -9.43 -8.93 15.09 13.70 11.86 11.20 10.77 28.20 37.15 46.35 64.10 76.75 14.80 17.20 17.65 21.80 24.5530 -8.55 -7.35 -6.10 -5.58 -5.35 11.37 9.59 7.66 6.79 6.18 25.60 29.35 35.60 49.60 71.00 10.65 9.95 8.95 10.10 10.3050 -6.39 -4.86 -3.74 -3.05 -3.19 8.82 6.71 4.87 3.77 3.81 26.60 26.60 28.30 36.55 59.50 10.95 10.25 10.05 22.35 31.90100 -4.42 -3.23 -1.98 -1.61 -1.56 6.14 4.68 2.89 2.10 1.87 26.80 25.60 19.70 24.80 38.70 12.65 19.40 40.40 73.05 93.65200 -3.57 -2.37 -1.21 -0.84 -0.78 5.19 3.57 1.93 1.30 1.03 32.05 25.30 16.60 17.35 24.10 24.55 42.80 79.25 98.80 100.00


20 -31.24 -27.21 -23.95 -22.96 -22.95 32.64 28.30 24.70 23.46 23.37 89.70 92.75 96.35 99.60 100.00 78.30 82.20 85.40 95.90 98.2530 -25.74 -21.23 -18.28 -16.52 -16.52 26.93 22.12 18.86 16.89 16.81 90.55 93.60 97.60 99.60 100.00 78.20 76.50 82.60 90.75 97.6550 -20.76 -16.51 -13.40 -11.39 -10.92 21.63 17.17 13.81 11.69 11.12 94.65 95.85 98.35 99.85 100.00 78.65 73.60 70.95 73.80 86.75100 -16.31 -12.50 -9.58 -7.70 -6.67 16.92 12.94 9.90 7.88 6.81 96.60 97.65 98.25 99.80 99.95 79.25 68.50 52.60 40.60 32.25200 -14.51 -10.80 -8.05 -5.85 -4.57 14.98 11.13 8.28 5.98 4.66 99.50 99.65 99.30 99.85 99.95 83.90 69.50 47.75 14.70 13.00

PC2POOL20 -31.95 -27.52 -24.18 -23.50 -24.05 33.04 28.47 24.87 24.02 24.56 95.80 96.50 98.25 99.80 100.00 87.95 87.00 89.80 96.95 99.3030 -26.27 -21.47 -18.38 -16.67 -16.96 27.25 22.29 18.91 17.06 17.28 96.35 96.25 98.95 99.75 100.00 86.75 83.60 86.75 92.40 98.3550 -21.31 -16.46 -13.29 -11.34 -11.05 22.05 17.04 13.68 11.63 11.26 98.80 98.00 99.00 99.90 100.00 89.35 81.05 76.05 74.65 88.35100 -16.95 -12.65 -9.52 -7.62 -6.67 17.50 13.05 9.81 7.80 6.81 99.40 99.40 99.50 99.95 100.00 90.45 78.30 58.35 41.55 32.65200 -15.07 -10.92 -7.93 -5.72 -4.52 15.52 11.24 8.15 5.85 4.60 99.90 99.95 99.90 99.90 99.95 94.80 80.20 51.05 14.00 15.30

Notes: The DGP is the same as that of Table 1, except βij = 1 for all i and j, i = 1, 2, ..., N , j = 1, 2. See notes to Table 1.

Table 3: Small Sample Properties of Common Correlated Effects Type Estimatorsin the Case of Experiment 1B (Heterogeneous Slopes + Rank Deficient)


CCEMG20 0.33 -0.19 0.20 0.14 0.23 15.02 13.90 12.61 13.35 13.78 6.80 6.90 6.75 6.60 7.20 9.40 8.95 10.15 10.15 10.1530 0.30 0.14 0.09 -0.17 0.35 12.91 12.03 10.70 10.07 10.59 5.50 6.80 5.25 6.15 4.80 8.40 10.05 9.45 10.35 11.6550 -0.15 0.63 -0.20 -0.17 0.02 9.82 8.46 7.87 7.42 7.34 5.80 5.10 6.10 5.75 5.90 9.75 12.90 13.40 14.00 15.20100 0.25 0.13 0.27 0.00 0.06 7.01 6.55 5.85 5.25 5.01 5.75 5.95 5.45 5.45 6.10 14.50 17.75 21.65 22.65 27.30200 0.05 -0.11 -0.17 -0.07 -0.05 5.35 4.65 4.15 3.61 3.31 4.80 5.05 4.75 5.15 4.55 19.45 23.70 29.75 37.25 43.45

CCEP20 0.48 0.06 -0.04 0.16 0.10 13.13 12.81 12.21 13.57 15.30 6.75 7.40 7.00 6.65 6.75 9.90 10.20 10.40 10.35 10.2530 -0.23 -0.06 0.18 -0.25 0.43 11.48 10.70 10.39 9.95 11.04 6.10 6.90 5.70 6.00 5.50 9.05 9.95 10.55 10.25 10.6050 0.00 0.48 -0.18 -0.17 -0.02 8.42 7.57 7.23 7.22 7.22 5.25 5.90 6.25 5.30 5.50 11.40 14.05 14.15 14.35 15.20100 0.11 0.18 0.24 -0.06 0.05 5.87 5.72 5.27 4.87 4.98 5.10 6.00 5.40 4.95 6.00 17.25 19.60 23.50 23.55 27.00200 0.04 -0.10 -0.16 -0.04 -0.03 4.35 3.99 3.75 3.30 3.15 5.40 4.70 5.25 4.10 3.95 25.75 28.50 34.50 41.10 46.05


20 -8.33 -8.02 -7.93 -7.27 -7.44 14.50 12.28 10.98 9.59 9.25 13.35 16.10 21.45 24.50 28.80 7.20 7.35 7.20 7.90 8.0030 -5.36 -4.97 -4.70 -4.58 -4.27 10.67 8.66 7.30 6.53 5.96 10.00 11.45 13.80 17.60 18.75 5.85 5.40 5.35 5.40 5.6050 -3.15 -2.86 -2.99 -2.59 -2.63 7.44 5.94 5.15 4.33 4.07 7.95 8.25 12.35 11.65 14.30 6.40 6.50 8.70 12.45 11.50100 -1.35 -1.27 -1.33 -1.26 -1.17 5.04 4.01 3.24 2.68 2.55 6.80 8.00 7.85 7.40 8.40 13.50 18.90 24.85 31.90 42.30200 -0.85 -0.77 -0.72 -0.58 -0.62 3.56 2.83 2.18 1.83 1.64 4.95 6.35 6.05 6.75 6.30 23.40 36.25 53.50 72.15 80.20

PC1POOL20 -7.93 -7.85 -7.87 -7.23 -7.32 12.94 11.59 10.53 9.43 9.14 13.45 16.70 22.35 25.30 28.35 7.50 7.30 7.60 7.65 7.3530 -5.33 -4.94 -4.61 -4.54 -4.21 9.81 8.23 7.13 6.50 5.92 11.10 12.55 13.25 17.70 18.40 6.55 5.90 5.60 5.70 5.3550 -2.98 -2.78 -2.92 -2.63 -2.65 6.62 5.67 5.01 4.37 4.12 8.00 9.85 11.35 12.15 14.20 7.40 7.90 8.40 11.70 11.50100 -1.41 -1.23 -1.36 -1.31 -1.21 4.37 3.66 3.19 2.71 2.57 6.80 7.25 7.50 7.70 9.15 15.85 21.20 26.30 31.30 40.80200 -0.82 -0.77 -0.75 -0.60 -0.64 3.02 2.57 2.12 1.82 1.67 6.05 6.35 6.65 5.90 6.75 29.20 42.25 56.25 71.40 78.95


20 -30.74 -26.62 -23.99 -22.56 -23.18 32.50 27.94 25.06 23.36 23.92 82.50 87.55 90.45 95.10 97.55 70.50 72.50 74.75 81.15 87.8030 -24.89 -20.85 -18.19 -16.68 -16.43 26.28 21.96 19.03 17.35 17.03 84.15 86.05 90.35 93.60 96.05 67.80 65.45 66.75 69.85 75.6550 -19.61 -15.65 -13.13 -11.38 -10.85 20.69 16.50 13.80 11.93 11.34 87.30 86.20 87.95 89.65 91.70 65.10 57.00 51.20 45.00 46.35100 -15.19 -11.94 -9.58 -7.57 -6.60 15.96 12.53 10.09 7.96 7.03 91.80 90.95 88.90 87.35 82.65 65.05 50.60 37.45 18.45 14.20200 -13.64 -10.37 -7.98 -5.75 -4.58 14.16 10.80 8.33 6.05 4.85 98.50 96.95 95.00 90.60 83.65 72.00 55.40 31.15 9.30 6.80

PC2POOL20 -31.30 -27.08 -24.30 -23.16 -24.26 32.81 28.29 25.32 23.99 25.10 88.95 89.95 93.15 96.00 98.75 79.05 77.70 79.10 83.00 90.2530 -25.55 -21.11 -18.14 -16.84 -16.86 26.80 22.15 18.99 17.53 17.49 90.25 88.25 90.40 94.00 96.30 75.35 68.95 66.20 70.00 77.1550 -19.99 -15.74 -13.00 -11.35 -10.96 20.92 16.56 13.65 11.91 11.46 93.30 89.05 88.30 88.65 91.70 74.15 60.30 51.40 45.55 47.50100 -15.72 -11.98 -9.51 -7.52 -6.60 16.43 12.56 10.01 7.91 7.03 96.00 92.55 89.40 86.50 81.95 74.30 54.45 36.55 17.60 14.00200 -14.09 -10.47 -7.86 -5.63 -4.52 14.61 10.90 8.21 5.93 4.79 99.30 97.95 95.20 88.75 81.90 82.55 60.20 32.10 8.25 6.15

Notes: The DGP is the same as that of Table 1, except γi2 ∼ IIDN (0, 1), so that the rank condition is not satisfied. See notes to Table 1.

Table 4: Small Sample Properties of Common Correlated Effects Type Estimatorsin the Case of Experiment 2B (Homogeneous Slopes + Rank Deficient)


CCEMG20 -0.28 -0.26 0.41 -0.31 0.73 14.45 12.85 12.02 12.07 13.47 7.35 5.45 6.40 6.70 6.00 9.35 9.15 10.95 11.55 10.9030 -0.11 0.07 0.09 0.45 -0.05 11.99 10.78 9.82 9.52 10.33 5.20 5.90 5.95 6.50 6.55 7.85 10.50 12.40 14.35 14.9050 0.00 0.23 -0.07 -0.02 0.00 9.01 7.97 7.62 6.79 6.72 5.05 4.80 5.00 5.45 4.95 9.40 12.20 15.75 17.60 21.15100 0.14 -0.08 -0.12 -0.03 0.06 6.66 5.92 5.16 4.78 4.56 4.65 5.40 5.60 4.60 6.35 15.10 18.15 23.95 28.50 34.85200 0.14 0.11 0.01 -0.17 -0.07 5.13 4.45 3.88 3.27 3.34 5.45 5.10 5.45 4.65 5.15 22.35 28.80 36.60 44.75 56.70

CCEP20 -0.12 -0.19 0.35 -0.26 0.66 12.66 11.53 11.56 12.12 15.07 7.45 7.00 7.55 6.35 6.50 9.85 10.00 12.60 12.65 11.5030 -0.09 0.05 0.06 0.39 0.03 10.00 9.57 9.26 9.36 11.05 5.55 5.75 6.80 6.70 6.75 9.90 11.70 13.30 15.20 14.5050 -0.14 0.39 -0.08 0.01 0.03 7.29 6.92 6.84 6.58 6.79 4.95 5.25 5.45 5.60 4.85 11.25 15.60 16.65 19.95 20.40100 0.20 -0.13 -0.11 -0.05 0.04 5.44 4.97 4.55 4.45 4.39 4.80 5.35 5.40 4.95 6.05 20.60 22.65 28.35 31.40 36.80200 0.19 0.11 -0.08 -0.13 -0.07 3.97 3.71 3.35 2.96 3.09 5.25 5.15 5.05 5.00 5.60 31.95 38.45 44.30 50.70 60.40


20 -8.22 -8.36 -7.93 -7.89 -7.48 13.75 11.78 10.07 9.07 8.40 13.30 18.95 27.70 46.00 58.60 7.25 7.80 8.20 9.55 9.4030 -5.40 -5.01 -4.70 -4.59 -4.52 10.11 8.07 6.49 5.54 5.03 10.35 13.45 19.70 33.45 52.05 5.95 5.45 6.45 5.80 4.9550 -3.24 -3.09 -2.81 -2.66 -2.66 6.96 5.35 4.22 3.39 3.04 9.40 11.05 15.90 25.40 44.85 6.85 7.80 11.60 22.45 35.50100 -1.53 -1.45 -1.28 -1.33 -1.28 4.54 3.40 2.57 1.96 1.64 6.10 7.75 10.25 15.80 26.30 13.10 20.95 41.60 71.75 95.70200 -0.80 -0.76 -0.67 -0.64 -0.63 3.40 2.39 1.74 1.22 0.94 6.00 5.45 7.75 10.75 15.70 27.15 46.80 79.60 98.65 100.00

PC1POOL20 -7.56 -7.83 -7.57 -7.61 -7.12 11.45 10.60 9.32 8.59 7.81 14.05 21.70 31.20 48.25 57.85 6.40 8.35 7.55 8.90 8.3530 -5.00 -4.76 -4.50 -4.43 -4.35 8.47 7.20 6.04 5.25 4.82 11.70 15.75 21.45 35.10 51.40 6.00 5.70 6.50 5.65 5.4050 -2.89 -2.77 -2.75 -2.60 -2.61 5.66 4.63 3.92 3.26 2.97 8.90 11.55 17.50 27.45 44.70 7.85 9.60 12.95 24.65 38.05100 -1.38 -1.39 -1.26 -1.31 -1.27 3.67 2.95 2.29 1.87 1.60 7.80 8.70 10.50 17.90 28.50 19.90 28.70 51.45 79.70 97.35200 -0.72 -0.74 -0.67 -0.63 -0.63 2.54 1.93 1.55 1.14 0.91 6.25 5.90 9.00 11.80 17.10 43.25 64.40 89.05 99.45 100.00


20 -30.58 -26.66 -23.75 -22.85 -23.07 32.14 27.74 24.51 23.36 23.51 87.40 91.15 95.95 99.60 99.95 75.80 79.65 85.50 94.95 98.8530 -25.10 -20.76 -18.15 -16.58 -16.60 26.40 21.70 18.75 16.97 16.90 89.25 92.45 97.20 99.75 100.00 74.35 74.15 80.85 90.90 98.1050 -19.66 -15.83 -13.16 -11.36 -10.91 20.58 16.50 13.60 11.67 11.12 92.45 94.35 97.90 99.75 100.00 72.90 69.30 69.35 72.45 86.90100 -15.37 -11.97 -9.44 -7.65 -6.66 16.01 12.43 9.75 7.84 6.80 95.30 96.70 98.00 99.70 99.95 74.25 63.40 50.10 40.00 32.15200 -13.52 -10.36 -7.93 -5.82 -4.57 14.01 10.70 8.17 5.96 4.66 98.80 99.15 99.40 99.90 99.95 76.85 63.50 45.30 14.80 13.35

PC2POOL20 -31.34 -27.06 -24.05 -23.47 -24.10 32.52 28.01 24.76 24.00 24.63 95.25 95.90 97.85 99.95 100.00 86.20 86.30 89.00 97.50 99.4030 -25.58 -20.90 -18.28 -16.76 -17.05 26.61 21.72 18.83 17.15 17.38 95.30 95.80 98.85 99.85 100.00 84.80 81.60 85.35 93.05 98.5050 -20.12 -15.76 -13.06 -11.33 -11.05 20.90 16.33 13.46 11.63 11.26 97.65 97.50 98.90 99.75 100.00 85.10 77.55 73.60 75.00 88.75100 -15.88 -12.09 -9.36 -7.58 -6.66 16.46 12.51 9.65 7.76 6.80 99.10 99.00 99.45 99.85 99.95 86.15 72.45 55.15 39.85 32.95200 -13.95 -10.45 -7.81 -5.69 -4.52 14.41 10.78 8.03 5.82 4.60 99.70 99.85 99.75 99.85 100.00 90.10 74.75 49.05 13.20 15.30

Notes: The DGP is the same as that of Table 1, except γi2 ∼ IIDN (0, 1), so that the rank condition is not satisfied, and βij = 1 for all i and j, i = 1, 2, ..., N , j = 1, 2. See notes to Table1.

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